Thermodynamics and Phase Coexistence in Nonequilibrium Steady States

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1 Thermodynamics and Phase Coexistence in Nonequilibrium Steady States Ronald Dickman Universidade Federal de Minas Gerais R.D. & R. Motai, Phys Rev E 89, (2014) R.D., Phys Rev E 90, (2014) R.D., New J Phys (2016) SUPPORT: CNPq, Brazil

2 Equilibrium Nonequilibrium

3 Nonequilibrium steady state (NESS): Macroscopic properties are time-independent But there is a steady current (of particles, energy,...) Focus on far-from-equilibrium systems NESS is closest analog to equilibrium, in far-from-equilibrium context

4 NESS

5 Key questions: Is there a thermodynamics of nonequilibrium steady states? What is a nonequilibrium phase? Is there a nonequilibrium entropy?

6 OUTLINE I. Steady-state thermodynamics (SST) Definition of intensive variables (chemical potential and temperature) via coexistence with a reservoir Works in spatially uniform systems if we use Sasa-Tasaki transition rates (otherwise, zeroth-law violations) Fails in nonuniform systems II. Phase coexistence of NESS Phase coexistence ill-defined, SST doesn't predict coexisting phases III. Nonequilibrium entropy is not the Shannon entropy

7 SST originally proposed in Y. Oono and M. Paniconi, Prog. Th. Phys. Supp. 130, 29 (1998) A detailed proposal with applications to lattice gases: S. Sasa and H. Tasaki, J. Stat. Phys. 125, 125 (2006) Numerical test of zeroth law in KLS driven lattice gas: P. Pradhan, R. Ramsperger, and U. Seifert, Phys. Rev. E84, (2011)

8 Easiest case: spatially uniform, athermal (only excludedvolume interactions) In athermal systems the only intensive parameter is the dimensionless chemical potential, m* = m /kbt Criteria for a valid chemical potential: 1) (Zeroth law) If systems A and B coexist, and A and C coexist, then B and C must also coexist 2) Suppose systems A and B, initially isolated, with m*a m* B. If they are allowed to exchange particles, the ensuing flux should act to equalize their chemical potentials. Knowing m*a(r) and m* B(r) should allow us to predict the final densities when the systems are allowed to exchange particles.

9 Coexistence of NESS Consider systems A and B in steady states (equilibrium or not) If A and B can exchange particles, and the net particle flux between them is zero, we say they coexist wrt particle exchange; we declare them to have the same value of m* Analogous definition of temperature for coexistence wrt energy exchange If B is a particle reservoir of known chemical potential the zero-flux condition defines the chemical potential of A

10 What is a reservoir? - A system large enough that its intensive properties don't vary under exchange of particles and/or energy - Internal structure irrelevant - The reservoir doesn't know what kind of system it's in contact with, in particular, whether this system is in equilibrium or not. A particle reservoir R in contact with an athermal lattice gas A attempts to insert particles into, and remove particles from A, with rates pi and pr, respectively. In insertion attempt is successful if the selected site is free to be occupied; a removal attempt is successful if the selected site is occupied. From equilibrium statistical mechanics we know pi /pr = e m* m* is a property of the reservoir

11 Application to driven lattice gas with nearest-neighbor exclusion (NNE) Each particle excludes nearest-neighbor sites hard-core repulsive potential Equilibrium dynamics: symmetric nearest-neighbor hopping attempts Nonequilibrium (driven) dynamics: preferred hopping direction Can include hopping to second-neighbor sites to maintain ergodicity at higher densities

12 NNE Lattice Gas

13 NNE Lattice Gas on square lattice: hopping probabilities for nearest-neighbor dynamics 1/4 1+D 4 1 D 4 1/4 Direction of drive: periodic boundaries along this direction For D=0 the dynamics obeys detailed balance and the stationary distribution is that of equilibrium D 0 represents nonequilibrium: the is a current along drive D=1: maximum drive

14 NNE lattice gas and other athermal systems: chemical potential given by m* = ln (r /r op) where r op is the density of open sites (sites at which particles may be inserted) This follows from the condition of coexistence with the particle reservoir: r pr = rop pi and the relation pi /pr = e m*

15 driven pr undriven Global exchange between driven and undriven systems Weak exchange limit pr 0 Obs: Global exchange not strictly necessary; weak exchange is

16 Zeroth law: If systems A and B both coexist with the same reservoir, they must coexist with one another The zeroth law is satisfied under weak, global exchange: the net particle flux between two systems A and B is proportional to ra rop,b - rb rop,a, which is zero if m*a = m*b Note that we treat the systems as independent, which implies an exchange rate tending to zero ( weak exchange)

17 Chemical potential of NNE lattice gas D=0 D=1

18 NNE lattice gas: weak-exchange limit

19 Driven NNE lattice gas In the weak-exchange limit, the chemical potential at coexistence corresponds to that of the isolated systems at the densities that equalize their m* values, subject to the fixed total density. Since m* is an increasing function of density, if NNE models with different values of m* are permitted to exchange particles, the ensuing flux will tend to equalize the chemical potential. Thus m* satisfies the minimal conditions for a chemical potential, both in equilibrium and in a NESS.

20 Driven lattice gas or Katz-Lebowitz-Spohn (KLS) model with attractive nearest-neighbor interactions. The system evolves via a particle-conserving dynamics with a drive D = Di favoring particle displacements along the +x direction and inhibiting those in the opposite sense. ***PBCs along drive! Acceptance probability for a particle displacement Δx is pa = min{1, exp[ (ΔE D Δx)/TR ]} Energy of configuration C: E(C) = - S<i,j> sisj The sj are site occupation variables TR is the temperature of the reservoir, but (for D 0) not the temperature of the system

21 Energy flow though KLS system Heat Work by drive KLS system Reservoir at temperature Tn

22 We define the temperature and chemical potential of a KLS system (driven or not) via coexistence with a reservoir R of heat and particles R is a test system used to measure the temperature and chemical potential of the system, not the reservoir to which the KLS system rejects heat. The reservoir T and m are defined by the usual relations of statistical mechanics. When the KLS system coexists with R it shares its values of T and m. [In paractice T and m are determined by measuring certain nearestneighbor probabilities in simulations of the system; explicit contact with a reservoir is unnecessary to determine the coexistence parameters.]

23 Let R be a reservoir with temperature and chemical potential equal to that of a driven KLS system S, and let S0 be an undriven (equilibrium) nearest-neighbor lattice gas with the same temperature and chemical potential as R and S. If the Zeroth Law holds, S and S0 must coexist. Do they? No! Although S and S0 both coexist with R, in general, they do not coexist with each other, violating the zeroth law The Zeroth Law does hold for Sasa-Tasaki exchange rates between the systems Numerical studies (simulation and exact solution of master eq) yield the fluxes between S and S0

24 fluxes Zeroth law violations under Metropolis exchange rates Equivalent to a temperature mismatch of ~0.4%

25 Sasa-Tasaki (ST) rates J Stat Phys 125, 125 (2006) The rate for transferring a particle from A to B depends only on the energy change and parameters of A, and vice-versa. Detailed balance implies the rate of particle transfer from A to B: WST = e exp [-(m A + D EA)/TR,A ] and similarly for B to A. (e is an arbitrary fixed rate.) Physical picture: rate is determined by energy barrier to activated state, not energy difference between initial and final states, as in, e.g., Metropolis rates

26 KLS model: Effective temperature and chemical potential under ST exchange (reservoir temperature TR=1, D=10)

27 Summary of results on uniform KLS lattice gas The intensive parameters T and m are defined via coexistence with a reservoir R. For drive D>0, T (=TR) > TR Only ST rates permit a consistent definition of intensive parameters for NESS We appear to have a general scheme for defining the temperature and chemical potential of spatially uniform systems in a NESS

28 Nonuniform athermal systems Given the success of steady-state thermodynamics using ST rates, applied to spatially uniform systems, we turn to NESS with nonuniformities provoked by: - a nonuniform drive - a wall - nonuniform time scale Steady-state thermodynamics fails in these cases

29 Test of SST in half-driven NNE lattice gases (ST rates): Can SST predict coexisting densities in a nonuniform system? Drive No drive

30 Half-driven NNE model For the same density, m* is smaller in the driven system than without drive. Equating chemical potentials, the density in the driven region should be larger. Nevertheless, the particles migrate in the opposite sense!

31 Stationary density and chemical potential profiles in half-driven NNE predicted observed undriven driven observed predicted

32 Conclusions (I) We appear to have a general scheme for defining the temperature and chemical potential of spatially uniform systems in a NESS A pair of systems (driven/undriven) that coexist under weak exchange don't coexist when in contact along an edge This result doesn't depend on whether we define a chemical potential m In half-driven system particles migrate contrary to Fick's law: the final state has a larger m difference than the initial uniform state

33 II. Can SST predict the densities of coexisting phases? KLS model - under drive, phase separation into strips oriented along drive; - use Sasa-Tasaki rates for particle movement perpendiclar to drive: necessary for consistency of SST under coexistence between rows parallel to drive

34 KLS: Typical phase-separated configuration, TR = 0.6 Drive

35 Phase coexistence: between two phases in a single system or between two single-phase systems Equilibrium rv Coexisting densities are the same in either case Equilibrium: a phase is a phase is a phase rl NESS We expect the coexisting densities to be the same, but are they?

36 Phase coexistence in KLS model Study phase separation in a single system under maximum drive, to identify coexisting densities at a given temperature TR, well below Tc For the same temperature, determine the coexisting densities for a pair of uniform systems under weak exchange If notions of phase and phase coexistence are applicable to the NESS, the coexisting densities must be the same in the two cases Results for TR=0.5: In phase-separated single system, coexisting densities are (4) and (1) In uniform systems under weak exchange, coexisting densities are (3) and (1) - clearly incompatible w/prev result - similar findings at other temperatures

37 Coexisting densities in single system and in coexisiting uniform systems: KLS model, maximum drive two systems single system

38 Is the lack of well defined coexisting phases due to the drive/current in the KLS model? To answer this I study another model, a two-temperature lattice gas: Sites on different sublattices (A/B) are in contact with different heat reservoirs (TA/TB) There is no drive, no particle current, no anisotropy The coexisting densities again depend on how the phases make contact For TA=0.4 and TB = 0.55, the coexisting densities are: (3) and (3) (single system) (6) and (6) (two systems, weak global exchange)

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40 Conclusions (II) Coexistence between systems in nonequilibrium steady states is not well defined: it depends on the manner in which the systems make contact. The properties of coexisting phases depend on how they exchange particles between one another. The notion of phase as a system with well defined intensive properties seems not to apply in NESS Since this inconsistency is found in two of the simplest possible cases, independent of drive or anisotropy, there is good reason to believe it occurs in general It appears unlikely that any formulation of steady-state thermodynamics can predict the properties of nonuniform systems or phase coexistence

41 III. In the spatially uniform case, the chemical potential is well defined and predictive: Can we define an entropy function? The thermodynamic entropy has the property understood as a finite difference for small systems. Thus we can evaluate S via integration. Recalling the relation we have S(N,L) S(N-1,L) = On a lattice of Ld sites. This defines the thermodynamic entropy.

42 Is the thermodynamic entropy for a NESS the Shannon entropy? SS = -kb Si pi ln pi To evaluate SS we need the stationary probability distribution, pi In collaboration with my student Leonardo Ferreira Calazans, we determine pi for the driven NNE lattice gas on square lattices of LxL sites. The results show that Sthermo SS

43 equilibrium max drive

44 Difference between thermodynamic and Shannon entropies at low drives

45 Violation of equality between Shannon and thermodynamic entropies:

46 Violation of equality between Shannon and thermodynamic entropies:

47 NNE model: difference between thermodynamic and Shannon entropies L=4 L=7 No difference for D=0; difference grows with L and N; proportional to D2 for small D

48 Given the stationary probability distribution Pest(C), we evaluate the mean energy transfer between the KLS system and reservoir R, as a function of the reservoir temperture TR: The value of T such that JE = 0 defines the thermodynamic temperature of the KLS system. The thermodynamic entropy is found via: with Sth = SS in limit T (at infinite temperature, drive is irrelevant)

49 Sth and SS in KLS model Sth SS

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51 Who cares if Sthermo SS? 1. The two are always equal in equilibrium 2. The Shannon entropy is widely used as the entropy out of equilibrium (e.g., in stochastic thermodynamics) 3. The Shannon entropy is needed to connect thermodynamics with information 4. The Shannon entropy is the only extensive functional of the probability distribution Since the thermodynamic entropy of a NESS is extensive (we verify that the chemical potential is intensive), this implies that Sthermo cannot be written as a functional of pi

52 Thank you!

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